There are three specific types of chains that are of interest: Cunningham chains of the first kind, Cunningham chains of the second kind, and “bi-twin” chains. The rule behind a Cunningham chain of the first kind is that each prime in the chain must be one less than twice the previous. The first Cunningham chain of length 5, for example, consists of the following six primes:

1531, 3061, 6121, 12241, 24481

In Cunningham chains of the second kind, each prime must be one more than twice the previous. Here, the first length-5 chain appears much sooner:

2, 5, 11, 23, 47

Finally, bi-twin chains are chains of pairs of twin primes, or primes that are 2 units apart from each other, with the average of each pair being twice the average of the previous pair. Each bi-twin chain must obviously have even length; the first chain six primes long is:

211049, 211051, 422099, 422101, 844199, 844201

Why is solving primes important? Bauterin explains:

searching for primes leads to useful byproducts in other areas of number theory, provides an incentive for computational hardware development and leads to insights in the underlying workings of prime numbers themselves

Primecoin would have a long way to catch up with Bitcoin, but things move fast in the crypto-currency world, so anything could happen.